Threshold behavior.
While we do not have a formal characterization here, we observe that the optimal strategy exhibits some threshold behavior, particularly near the end of the liquidation.
We observe that the policy trades aggressively when the cumulative cost \(C_t\)  is below some threshold, and it trades passively when the cumulative cost \(C_t\)  is above the threshold.
Near the end of the liquidation, such a threshold corresponds to \(\operatorname{\textup {VaR}}_q[C_\infty ]\)  (the value-at-risk, i.e., the \(q^\text{th}\)  quantile of the loss distribution), and the liquidation rate sharply changes around \(\operatorname{\textup {VaR}}_q[C_\infty ]\) .
This behavior is related to an alternative representation of CVaR [1]}: \(\operatorname{\mathrm {CVaR}}_q[ C_\infty ] = \max _{c \in \mathbb {R}}\lbrace  c + \frac{1}{q} \mathbb {E}\left[ (C_\infty - c)^+ \right] \rbrace \) , where the maximizer \(c^\star \)  is in fact \(\operatorname{\textup {VaR}}_q[C_\infty ]\) , and \(\operatorname{\mathrm {CVaR}}_q[ C_\infty ]\)  only concerns the cases where \(C_\infty > c^\star \) .
